Question

Radical Expressions Evaluate the radical expression, and express the result in the form $a+bi$.

$\sqrt{\frac{1}{3}}\sqrt{-27}$

Short answer

The value of radical expression is $0+3i$.

Step-by-step solution

Step 1. Concept of complex number.

A complex number is of the form $a+bi$. Where a and b are some real numbers and i is the imaginary number.

The number a is known as the real part of the complex number and b is known as imaginary part of the complex number.

Step 2. Concept of imaginary number.

The imaginary number i is defined as the square root of negative one. That is, $i=\sqrt{-1}$or $i^2=-1$.

The properties of radicals,

• $\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$
  
• $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$
  

Step 3. Evaluate the radical expression.

To evaluate the given radical expression, use the concept of imaginary number and simplify further as shown below

$$\begin{gathered} \sqrt{\frac{1}{3}}\sqrt{-27}=\sqrt{\frac{1}{3}}\sqrt{-1\times 27} \\ =\sqrt{\frac{1}{3}}\times \sqrt{-1}\times \sqrt{27} \\ =\sqrt{\frac{1}{3}}\times i\times \sqrt{27} \\ =\sqrt{\frac{1}{3}\times 27}\times i \\ =\sqrt{9}\times i \\ =3\times i \\ =0+3i \end{gathered}$$

Thus, the value of radical expression is $0+3i$.